[Paper Review] Topological Borsuk problem

Yan Soibelman|ArXiv.org|Aug 28, 2002

Limits and Structures in Graph Theory1 references3 citations

TL;DR

This paper investigates the topological Borsuk number of Euclidean space ℝⁿ, defined as the minimal Borsuk number over all metrics topologically equivalent to the standard Euclidean metric. Using equivariant topology and the G-index for ℤ/3-action, it proves that B(ℝ²) = 3, establishing the topological Borsuk number for the plane. The method fails for n > 2 due to non-free group actions on target spaces, leaving the general case open.

ABSTRACT

Classical Borsuk problem asks about the minimal number of closed subsets of smaller diameter necessary to partition every compact in the Euclidean space. Topological version of the Borsuk problem is discussed.

Motivation & Objective

• To determine the topological Borsuk number B(ℝⁿ), defined as the infimum of B(X,ρ) over all metrics ρ topologically equivalent to the Euclidean metric.
• To investigate whether B(ℝⁿ) ≥ n+1, extending the classical Borsuk conjecture to a topological setting.
• To explore the role of group actions and equivariant topology in estimating Borsuk numbers.
• To identify obstructions preventing the 2D proof technique from generalizing to higher dimensions.

Proposed method

• Define the topological Borsuk number B(X) as the minimum Borsuk number over all metrics in the set Ω(ρ₀) of ρ₀-equivalent topologies.
• Use a continuous map f: X³ → ℝ³ that assigns pairwise distances to triples of points in ℝ².
• Prove the existence of an equilateral triple (ρ₁₂ = ρ₂₃ = ρ₁₃ > 0) in the image of f by showing no ℤ/3-equivariant map exists from S³ to S¹.
• Apply the G-index (group index) for G = ℤ/3, showing that ind_G(S³) = 3 > 1 = ind_G(S¹), implying no such equivariant map.
• Leverage the fact that odd-dimensional spheres admit free ℤ/p-actions for odd primes p, using p = 3 for the 2D case.
• Generalize the approach to higher dimensions by considering Σ_{n+1}-equivariant maps on (ℝⁿ)^{n+1}, but note that group actions become non-free on target spaces.

Experimental results

Research questions

• RQ1 Is the topological Borsuk number B(ℝⁿ) bounded below by n+1 for all n ≥ 2?
• RQ2 Can the 2D proof technique using ℤ/3-equivariance and G-index be extended to ℝⁿ for n > 2?
• RQ3 What topological obstructions arise when group actions on target spaces fail to be free in higher dimensions?
• RQ4 Does the existence of equilateral configurations in all ρ₀-equivalent metrics imply a lower bound on B(ℝⁿ)?
• RQ5 Are there metrics in Ω(ρ₀) for ℝⁿ, n > 2, where B(X,ρ) < B(ℝⁿ,ρ₀)?
• RQ6 key_findings
• RQ7 The topological Borsuk number of ℝ² is exactly 3, proven via equivariant topology and the G-index of ℤ/3-actions.
• RQ8 The proof relies on showing no ℤ/3-equivariant map exists from S³ to S¹, using the inequality ind_G(S³) = 3 > 1 = ind_G(S¹).
• RQ9 The standard Euclidean metric on ℝ² has Borsuk number 3, and this value is minimal over all topologically equivalent metrics.
• RQ10 The method fails in higher dimensions because the symmetric group Σ_{n+1} acts non-freely on the target space in the generalized map construction.
• RQ11 For ℝ², the existence of equilateral triples in any ρ₀-equivalent metric is guaranteed by the topological obstruction, confirming B(ℝ²) = 3.
• RQ12 The general case for ℝⁿ, n > 2 remains open, as the topological tools used in dimension 2 do not extend due to non-free group actions.

Key findings

• The topological Borsuk number of ℝ² is exactly 3, proven via equivariant topology and the G-index of ℤ/3-actions.
• The proof relies on showing no ℤ/3-equivariant map exists from S³ to S¹, using the inequality ind_G(S³) = 3 > 1 = ind_G(S¹).
• The standard Euclidean metric on ℝ² has Borsuk number 3, and this value is minimal over all topologically equivalent metrics.
• The method fails in higher dimensions because the symmetric group Σ_{n+1} acts non-freely on the target space in the generalized map construction.
• For ℝ², the existence of equilateral triples in any ρ₀-equivalent metric is guaranteed by the topological obstruction, confirming B(ℝ²) = 3.
• The general case for ℝⁿ, n > 2 remains open, as the topological tools used in dimension 2 do not extend due to non-free group actions.